**-1**

votes

**2**answers

142 views

### Motivation for weak solution of a PDE (initial condition)

The following question came to me when reading the famous paper of ALT and LUCKHAUS: "Quasilinear elliptic-parabolic differential equations"
When looking at a (nonlinear degenerate) PDE like
$$ ...

**4**

votes

**1**answer

510 views

### Questions on the proof of the Serrin condition for the regularity of Navier-Stokes equations and related issues for the incompressible Euler equation

Edit: The question has been substantially modified from the original one. The original question (see below) concerned with rigorously justifying the proof of the Serrin condition. These questions have ...

**0**

votes

**1**answer

61 views

### Reference for holder estimate on parabolic equation with neumann boundary condition

I saw a type of holder estimate in Friedman's book: partial differential equations of parabolic type(page 200 3.24) as following:
Suppose we have a uniformly parabolic equation with holder ...

**5**

votes

**0**answers

94 views

### Reference request: Optimal $L^p$-decay for nonhomogenous heat equation in $\mathbb R^d$

Let $u$ be a classical solution for the nonhomogeneous heat equation in $\mathbb R_+ \times\mathbb R^d$:
$$
\begin{cases}
\partial_tu(t,x)-\Delta u(t,x) = f(t,x), \\
u(0,x)=u_0(x).
\end{cases}
$$
...

**4**

votes

**0**answers

82 views

### Reference for short time existence of paraobolic PDE on bundles

I am looking for a reference treating parabolic equations on vector bundles. In particular, I look for precise conditions which guarantee short time existence. I found it quoted at different places in ...

**4**

votes

**0**answers

110 views

### Asymptotic higher order derivative estimates for solutions of semi-linear parabolic PDEs

Question:
Consider a semi-linear parabolic equation on a bounded domain $\Omega \subset \mathbb{R}^m$:
$$
\frac{\partial f_t}{\partial t} = -\Delta f_t + Q(f_t, df_t),
$$
with smooth initial data ...

**3**

votes

**0**answers

57 views

### Reference for existence results for 2D forced viscous Burgers equation

I am looking for results concering the following parabolic PDE
$$u\cdot\nabla u + \Delta u = F(x),$$
where $$u\colon\Omega\to\mathbb{R}^2,$$ and $\Omega\subset\mathbb{R}^2$ is a 2D domain (bounded or ...

**3**

votes

**0**answers

73 views

### Hopf Lemma for strong solutions

The Hopf lemma still holds for strong solutions? Let $u\in W^{2,p}(\Omega)\cap W_0^{1,p}(\Omega)$ with $p>n$ such that $\Delta u + c(x) u\le 0$. If $u(x)>0$ for all $x\in\Omega$ and ...

**3**

votes

**0**answers

149 views

### Need a regularity result for parabolic PDE, want $u' \in L^\infty((0,T)\times \Omega)$

Let us assume $\Omega \subset \mathbb{R}^n$ is as nice as required.
Let $f \in L^\infty((0,T)\times \Omega)$ and let $g \in L^\infty((0,T)\times \Omega)$ satisfy
$$0 < a \leq g(x,t) \leq b < ...

**2**

votes

**0**answers

48 views

### Properties of solutions of Parabolic type equations

Assume $u\in C([0,1],L^{2})$ satisfies the following Schrodinger equation
$$
\partial_t u=i(\Delta u+Vu), \text{in} ~\mathbb{R}^n\times[0,1],\\
u(0)=u_{0}.
$$
with $V=V_1(x)+V_2(x,t)$, where $V_{1}$ ...

**2**

votes

**0**answers

90 views

### Regularity of solution to Fokker Planck equation

Suppose that $\rho \in L^1(\mathbb{R}^n \times (0,T))$ for every $T < \infty$ is a weak solution of the PDE
\begin{align}
\partial_t\rho &= \Delta \rho + \text{div}(\rho\nabla\Psi(x))\\
\rho(t ...

**2**

votes

**0**answers

48 views

### Solve a PDE related to free boundary problem

I would like to solve the following system for my problem:
$$\max\Big(\frac{1}{2}u_{ss}+u_l\delta(s-s_0), F(l)-\lambda(s)-u(s,l)\Big)=0.$$
where $u=u(s,l): R\times R_+\to R$ is the unknown function ...

**2**

votes

**0**answers

79 views

### Weighted energy estimate for the heat equation of higher order

The question is originally related to Hardy's uncertainty principle, convexity and Schrodinger evolutions. In this work the authors deduce a convex property of Schrodinger equation by doing it first ...

**2**

votes

**0**answers

63 views

### Changing the test function space in a weak formulation of parabolic PDE

Suppose we are interested in the existence of a $u \in L^2(0,T;V)$ with $u' \in L^2(0,T;V^*)$ such that
$$(u(T),\varphi(t))_H -\int_0^T \langle \varphi'(t), u(t) \rangle_{V^*,V} + \int_0^T ...

**2**

votes

**0**answers

76 views

### Stabilization of solution to one-dimensional system of PDE

I am trying to solve numerically next PDE system:
$$\frac{\partial c}{\partial t}=\epsilon\frac{\partial}{\partial x}(\frac{\partial c}{\partial x}+\rho\frac{\partial \varphi}{\partial ...

**2**

votes

**0**answers

108 views

### Examples of non-uniqueness in reaction-diffusion equations

Consider the problem of finding a bounded classical solution $u:\mathbb{R}\times [0,T]\to\mathbb{R}$ (such that $u$ is continuous and $u_t$, $u_x$ and $u_{xx}$ exist and are continuous on ...

**2**

votes

**0**answers

105 views

### Existence of solutions to a reaction-diffusion problem.

Consider the problem of finding a bounded classical solution $u:\mathbb{R}\times [0,T]\to\mathbb{R}$ (such that $u$ is continuous and $u_t$, $u_x$ and $u_{xx}$ exist and are continuous on ...

**2**

votes

**0**answers

133 views

### regularity for viscosity solutions of second order parabolic equations

I would like to know whether viscosity solutions to $u_{t} - F( D^{2} (u) ) = 0$ are $C^{1, \alpha}$ analogous to the elliptic case as in the book by Caffarelli and Cabre .
Here F is ...

**1**

vote

**0**answers

46 views

### Uniform bounds for a coupled parabolic system of PDE (linear)

Let $V=H^1(\Omega)$ and $H=L^2(\Omega)$ where $\Omega$ is a compact Riemannian manifold. Define $W = \{ w \in L^2(0,T;V) : w_t \in L^2(0,T;V^*)\}$.
Consider the system, with $u^\epsilon, v^\epsilon ...

**1**

vote

**0**answers

45 views

### References Request : Existence and Uniqueness for PDE which is “ALMOST (?)” Parabolic

I am doing a research using PDE which is a little different from the standard Parabolic type. The following is my case:
\begin{equation}
Lu = - \sum_{i, j = 1}^{N}a^{i,j}(x, t)u_{x_{i}x_{j}} + ...

**1**

vote

**0**answers

46 views

### Well-posedness of a certain linear Cauchy-problem

I am interested in solutions to the linear Cauchy problem
$$\Bigl(\frac{\partial^2}{\partial t^2} + a(t, x)\frac{\partial}{\partial t} + \sum_{j=1}^n b_j(t, x) \frac{\partial}{\partial x_j}\Bigr)u(t, ...

**1**

vote

**0**answers

60 views

### What's Known About the Green's Function to the 1D Diffusion Equation with Position-dependent Diffusion Coefficient?

Consider a one-dimensional diffusion equation
$$
C(x) \partial_t \Phi(t,x) = \partial_x^2 \Phi(t,x),
$$
on the interval $[0,1]$. The function $C(x)$ has a pole of order 1 at $x=0$ and a pole of finite ...

**1**

vote

**0**answers

116 views

### A parabolic PDE with Lipschitz nonlinearity, how to obtain well-posedness?

Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^n$ (or more generally a compact manifold). I'm interested in well-posedness (existence most importantly) of equations of the form
$$u_t(t) - ...

**1**

vote

**0**answers

106 views

### Weak periodic solution of parabolic PDE

Take
$$
u_t(t) + A(t)u(t) = f(t),
$$
$$
u(0) = u(T),
$$
where $A$ is an linear elliptic operator and the first equation is an equality in $L^2(0,T;V^*)$ for $V \subset H \subset V^*$ Hilbert triple. ...

**1**

vote

**0**answers

96 views

### Proving solution to linear parabolic PDE is convex with negative third derivative

I have a PDE in $g(y,t)$ of the form
\begin{equation}
a\frac{\partial^2g}{\partial y^2}y^2-b\frac{\partial g}{\partial y}y -rg + \frac{\partial g}{\partial t} - c = 0
\end{equation}
in which $a$, $b$, ...

**1**

vote

**0**answers

140 views

### Comparison principle for partial differential equation with singular coefficients

How (or if) a comparison principle works in the case of equations
singular at some point? For example, I am analyzing a partial
differential equation
$$
...

**1**

vote

**0**answers

76 views

### monotone parabolic systems, convex variational structure and Legendre transform

The context:
for my research I am currently looking at parabolic systems of the type
$$
\left\{
\begin{array}{ll}
\partial_t b(u)-\Delta u=0 \qquad & (t,x)\in \mathbb{R}^+\times\Omega\\
u=0 & ...

**1**

vote

**0**answers

178 views

### local existence for a singular quasilinear parabolic equation

I'm considering the following type of PDE:
$u_{t}=u_{xx}+u_{x}+u_{x}^2+u_{x}^3+\frac{u_{x}}{x(1-x)}+\left(\frac{u_{x}}{x(1-x)}\right)^3$
with periodic boundary conditions $u_{x}(0,t)=u_{x}(1,t)=0$, ...

**1**

vote

**0**answers

62 views

### Semiflows and continuous symmetries

Given a differential equation on a Banach space $\mathcal{X}$ of the form $\frac{d u}{d t} = F(u)$, it is often the case that $F$ is equivariant under translations, i.e. that $T_\alpha F(u) = ...

**0**

votes

**0**answers

37 views

### Reference for a regularity result for linear parabolic equations under mixed Neumann-Dirichlent boundary condition

Could someone please help me to find a precise reference for $L^{p}%
-$regularity results for linear parabolic equations under mixed
Neumann-Dirichlet conditions? More precisely, I would like to have ...

**0**

votes

**0**answers

74 views

### Heat equation with graph laplacian

I would like to start with considering the time-dependent heat equation on a connected graph and consider its Laplacian matrix.
Suppose we have a connected graph with unknown temperature on vertices. ...

**0**

votes

**0**answers

64 views

### Solving a parabolic PDE with boundary conditions given over ranges

How can one solve a Parabolic PDE (like the wave or diffusion equations) if the boundary conditions were given over ranges?
Here is an example: How to solve the equation ...

**0**

votes

**0**answers

82 views

### Heat equation with heterogeneous heat conduction

I'm trying to discretize and a heat conduction/diffusion problem using finite differences and I was wondering how to use a discrete heat conduction coefficient defined per cell (instead of per ...

**0**

votes

**0**answers

62 views

### Weak convergence of 4-th degrees

Good day!
We have an equation $y'+Ay=Bu$ where $y=\{\theta,\varphi\}$, $A, B$ are nonlinear operators.
$u \in L^\infty(\Gamma)$, $\theta, \varphi \in W = \{y \in L^2(0,T;V) : y'\in L^2(0,T;V')\}$, ...

**0**

votes

**0**answers

56 views

### Interpolation with time continuity

If $u(x,t)$ is a function depends on $x\in\Omega$ and $t\in[0,T]$. The following result could be found in L.C. Evans's book "PDE".
Suppose $u\in L^2(0,T;H_0^1(\Omega))$, with $u_t\in ...

**0**

votes

**0**answers

55 views

### Parabolic partial differential equation, initial conditions

Let $U\subset\mathbb{R}^n$ be open bounded, $T>0$.
Given the parabolic PDE $$\partial_tf+a\partial_xf+b\partial_{xx}f = g \qquad (1)$$ I'm interested in the initial and boundary conditions that ...

**0**

votes

**0**answers

242 views

### What is the solution of $u_t=u_{xx}+\frac{1}{x}u_x$?

What does the solution $u(x,t)$ of $u_t=u_{xx}+\frac{1}{x}u_x$ on $[0,1]$ with the following initial condition look like?
$u(\frac{1}{2},0)=\delta(\frac{1}{2})$ (i.e. delta function at ...

**0**

votes

**0**answers

154 views

### Smoothness of a solution for degenerate pde

I am looking at the IBVP:
$$
u_t+x^2u_{xx}+u_y+f(x,y)=0, x,y \in D\in [0,\infty)\times[0,\infty),t\in(0,T]\\
$$$$
u(x,y,0)=F(x,y)\in C^{1,1}\\
$$$$
u(x,y,t)=0, x,y\in \partial D
$$
and I would like to ...